A characterization of progressively equivalent probability measures preserving the structure of a compound mixed renewal process

Abstract

Generalizing earlier works of Delbaen & Haezendonck [5] as well as of [18] and [16] for given compound mixed renewal process S under a probability measure P , we characterize all those probability measures Q on the domain of P such that Q and P are progressively equivalent and S remains a compound mixed renewal process under Q with improved properties. As a consequence, we prove that any compound mixed renewal process can be converted into a compound mixed Poisson process through a change of measures. Applications related to the ruin problem and to the computation of premium calculation principles in an insurance market without arbitrage opportunities are discussed in [26] and [27], respectively.

A characterization of equivalent martingale probability measures in a mixed renewal risk model with applications in Risk Theory

“…(i) P P M ℓ S,KpΘq ; (ii) there exists a P Θ -null set L P P BpDq such that P θ P M ℓ S,Kpθq with pP θ q X 1 " P X 1 for any θ R L P . The validity of the above remark is an immediate consequence of [18], Proposition 3.3 along with the fact that X 1 P L ℓ pP q if and only if…”

“…For the definition of a P -conditionally (stochastically) independent process as well as of a P -conditionally identically distributed process over σpΘq we refer to [18]. We say that a process is P -conditionally (stochastically) independent or identically distributed given Θ, if it is conditionally independent or identically distributed over the σ-algebra σpΘq.…”

“…Ωq " BpΥ q N bBpΥ q N , Ω :" r ΩˆD and Σ :" r Σ bBpDq, let ℓ P t1, 2u, consider Θ, P P M ℓ S,KpΘq and tP θ u θPD as in [18], Proposition 4.1, put S pβq t pΘq :" ř Nt j"1 βpX j , Θq for any β P F ℓ P,Θ , and assume that E P rW 1 |Θs P Υ P ae σpΘq-a.s.. In the next result we give a characterization of all elements of M ˚,ℓ S,ExppρpΘqq in terms of the elements of F ℓ P,Θ ˆR˚,ℓ `pDq.…”

“…However, since the (mixed) Poisson risk model presents some serious deficiencies, as far as practical models are considered (see [14], p. 44 and [18], pp. 1-2), it seems reasonable to investigate the existence of EMMs for the price process U T in the more general mixed renewal risk model.…”

“…In [18], Corollary 4.8, a characterization of all progressively equivalent probability measures that convert a compound mixed renewal process CMRP for short) into a compound mixed Poisson one (CMPP for short) was proven. Relying on the above result, we provide the aforementioned characterization for a subclass of progressively equivalent measures, suitable for applications in Risk Theory, which improves the structure of a CMRP by converting it into a CMPP with better properties than those of [18], Corollary 4.8, see Theorem 3.5.…”